Testing for rational speculative bubbles in the Brazilian residential real-estate market

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by  Marcelo M. de Oliveira

Noname manuscript No.
(will be inserted by the editor)
Testing for rational speculative bubbles in the
Brazilian residential real-estate market
Marcelo M. de Oliveira · Alexandre C.
L. Almeida
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Received: date/Accepted: date
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Abstract Speculative bubbles have been occurring periodically in local or

] global real estate markets and are considered a potential cause of economic
N crises. In this context, the detection of explosive behaviors in the financial
G marketandthe implementation ofearly warningdiagnosistests areof critical
. importance.TherecentincreaseinBrazilianhousingpriceshasrisenconcerns
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that the Brazilian economy may have a speculative housing bubble. In the
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f presentpaper,weemploy arecentlyproposedrecursiveunit roottestin order
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q to identify possible speculative bubbles in data from the Brazilian residential
[ real-estate market. The empirical results show evidence for speculative price

bubbles both in Rio de Janeiro and Sa˜o Paulo, the two main Brazilian cities.
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Keywords Speculative Bubbles · Real-Estate Market · recursive unit-root
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tests · explosive behavior · price-rent ratio · Emerging markets
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7 JEL Classification: C15,C22,G01,R31.
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r Authorsafilliation:UniversidadeFederaldeS˜aoJoa˜odelRei,CampusAltoParopeba,
a 36420-000, OuroBranco,MinasGerais-Brazil
E-mail:[email protected](M.M.deOliveira)
E-mail:[email protected](A.C.L.Almeida)
2 MarceloM.deOliveira,AlexandreC.L.Almeida
1 Introduction
The efficient market hypothesis (EMH) assumes that all information is re-
flected instantaneously in assetprices [1]. Hence, marketprices should always
be consistent with the “fundamentals”. On the other hand, sudden dramatic
price changes over short periods of time, in which prices diverge from funda-
mentals, have been observedin severalmarkets around the world [2]. Famous
examplesincludetheDutch“Tulipmania”,the1929CrashintheUnitedStates
andtherecentAmericansubprimecrises[3,4].Suchbehaviorhasraisedmany
questions regarding market efficiency and has stimulated researchon rational
speculative bubbles [5,6].
Rationalspeculativebubblesoccurwhenthereisanexcessivepublicexpec-
tation of future price increasing, which produce rapid increases in valuations
of an asset [7]. For example, during a real-estate bubble, investors (homebuy-
ers) stay in the market despite the deviations of prices from fundamentals
becausethey expectsignificantfurther priceincreases.Additionally,whenthe
bubble is emerging, people think that real estate prices are very unlikely to
fall,lettingtoalittleperceivedriskassociatedwithinvestingintherealestate
market. But eventually the prices may reach unsustainable levels, and crash:
the bubble bursts [7,8,9].
Bubbles,rationalornot,havebeenoccurringperiodicallyinlocalorglobal
realestatemarketsandareconsideredofcriticalimportanceandafundamen-
tal cause of financial crises and ensuing economic crises [2,10,11,12]. Hence,
the study of real-estate bubbles is an important contribution to the literature
onfutureeconomicdevelopment.Moreover,abubbleishazardousforfinancial
and macro stability, since it can amplify a credit boom by inflating collateral
values and causing misallocation of economic resources. In order to minimize
the negative macroeconomic effects of a bubble, one needs to detect it early
as soon as prices begin to rise.
The usual definition of a bubble is a deviation of the asset price from
its fundamental value. However it can be very difficult to evaluate what is
the “fundamental” price.Alternatively, constantly diminishing dividend-price
ratiocanserveasareferenceforidentifyingarationalbubbleinastockmarket.
If price expectations are rising, but higher dividends fail to materialize, the
price rise is probably not based on fundamentals. A large number of studies
have been proposed to empirically detect asset bubbles, mostly focusing in
stock markets. Some of these methods have been applied to the study of real
estate bubbles. Econometrics tests, like unit root tests or cointegration tests
provide direct tests for the no-bubble hypothesis. If property price to income
(or rent) ratio is stationary or the property price is cointegrated with the
fundamental price, the no-bubble hypothesis cannot be rejected.
The first econometric tests for rational bubble detection were volatility
tests such as Schiller’s [13] variance-bound and West’s two-step test [14]. The
underlyingideaistocomparethevolatilitybetweentheassumedfundamental
asset price and the actual asset price. An asset bubble is detected indirectly
if the volatility of the actual assetprice is significantly largerthan that of the
TitleSuppressedDuetoExcessiveLength 3
assumed fundamental asset price. However, Marsh and Merton [15] pointed
evidence that variance bounds test fails when dividends and stock prices are
non-stationary.
Campbell and Shiller [16] and Diba and Grossman [17] introduced the
most commonly used methods for detecting asset price bubbles in the liter-
ature, namely the right-tailed unit root test and the cointegration test. The
aimofacointegrationtestistoexaminewhetherornotvariablesaretrending
together. If two series are cointegrated, that means the movements of these
two variablesarehighly correlated.Inother words,there is anequilibriumre-
lationshipbetweenthesetwoseries.Relatedtestsusingdatasetsthatcombine
timeseriesandcross-sections(Panel-basedtests)havebeenappliedrecentlyin
order to investigate the existence of real estate bubbles in Taiwan [18], China
[19], as well USA and Europe [20]. These methods, however,suffer from a se-
riouslimitation,firstpointed-outbyEvans[5],whoshowedthatcointegration
tests of asset prices and dividends are not able to detect explosive bubbles
when the sample data includes periodically collapsing bubbles.
Anovelapproachtoidentifyinganddatingbubblesinrealtimehasrecently
been introduced by Phillips and Yu [21]. Consideringthat the explosiveprop-
erty of bubbles is very different from random walk behavior, they developed
a recursive econometric methodology interpreting mildly explosive unit roots
as a hint for bubbles. The method provides a supremum Dickey-Fuller (DF)
test [22] that overcomes the problem identified with unit root and cointegra-
tion tests. The supremum DF test improves power significantly with respect
to the conventional unit root and cointegration tests, and has the advantage
of allowing estimation of the origination date and final date of a bubble. The
ideaistodetectspeculativebubblesastheyemerge,notjustaftertheirburst.
This methodology was generalized in [23] for detecting multiple bubbles. Yiu
etal[24]appliedthe latterfordetecting bubblesintheHongKongresidential
property market, Chen and Funke [25] for the Chinese housing market and
Gonz´alez et al. [26] for the Colombian housing market. A related approach
wasemployedbyKideval[27]fordetectingrationalbubblesintheUShousing
market.
It is noteworthy to mention a few tests for bubbles not based on econo-
metric approaches.Sornette et al.[28]intend to predictthe endofthe bubbles
assumingthat a crashfollowsafter rapidgrowthofeconomic indicatorsfaster
than an exponential function. Zhou and Sornette [29,30] examined whether
the price bubble burst in the US, and predicted that the turning point of the
bubblewouldoccuraroundmid-2006.Watanabeetal.[31,32]proposetoiden-
tify bubbles and crashes by exponential behaviors detected in the systematic
data analysis. More recently, Ohnishi et al. [33] observed that the land price
distribution in Tokyo had a power law tail during the bubble period in the
late 1980s,while it was very close to a lognormal before and after the bubble
period. This led them to argue that a characteristic of real estate bubbles is
not the rapid price hike itself but a rise in cross-sectionaldispersionof prices.
Overthepastfew decades,therealestatemarkethasbeenatargetofgov-
ernment fiscal and monetary policies aimed at achieving balanced economic
4 MarceloM.deOliveira,AlexandreC.L.Almeida
growth,lowunemploymentandlowinflation.Whenthehousingmarketorthe
overall economy is on a downturn, the governments tend to encourage banks
to adopt a lenient mortgage policy, increasing the available credit as well to
implement policies favorable to construction companies. Another usual gov-
ernment policy consists on subsiding disadvantaged groups through preferred
loan rates. As a consequence, the macroeconomic market is filled with specu-
lative capital demand and supply, resulting in inefficiency in the operation of
variousmarkets.This situationcreatesthe conditionsto theonsetofabubble
[18].
Such situation is exemplified by the Brazilian real estate policies. At the
end of 2007, the world was affected by the subprime mortgage crisis in the
United States. In response, Brazilian government established various market
stimulus policies [34], which includes: (i) Approval of a new law of fiduciary
alienationthatminimizestheriskfortheBrazilianbanksincasebuyersdefault
ontheir loans.This makethe banksmore willing to lendto potentially riskier
buyers. (ii) Brazil Growth Acceleration Program (referred to as PAC) which
included loans providedby government-ownedbanks,as well long-termcredit
forinfrastructure,sanitationimprovements,andmoreunderanewinvestment
fundthatbeganwith657,4billionBrazilianReals(BRL)1 inthe period2007-
2010 and more 955,1 billion BRL for the period 2011-2014 . (iii) Programs
administered by government-owned banks with the aim of development of
1,000,000 houses and apartments with subside for poor families, resulting in
a credit of 96 billion BRL in 2011, 18 times the amount of credit available in
2003.
In fact, real estate prices in Brazil have raised rapidly in the last few
years, and some analysts have been arguing the possibility that a bubble has
been inflated and could potentially burst [35,36,37,38,39]. In contrast,others
arguethatthe pricing growthis sustainableandbasedonfundamentals,since
Brazil was one of the fastest-growing major economies in the world in recent
years with an average annual Gross Domestic Product (GDP) growth rate
of over 5 percent, which made Brazilian economy the world’s seventh largest
by nominal GDP by the end of 2012. As property is a sizable component of
household and corporate balance sheets, a sudden collapse in property prices
mayhavenegativespillovereffectsontheoverallmacroeconomicsituationand
may pose macroeconomic and financial stability risks.
Therefore,theaimofthispaperistoinvestigatethebubble-likebehaviorof
the recent real estate market prices in Brazil. Identifying speculative bubbles
is not an easy task even in mature markets with long time series. Since in
Brazil the time series for house prices are short, the recent developed test by
Phillipsetal.[23],aimedatidentifyingexplosivebubblesinrealtime,provides
an adequate tool for such analysis [40].
1 Duringtheperiod2007-2012thevalueofUSdollar(USD)toBRLratewasintherange
1.55to2.46,withameanofabout2.00.Currently1.00BR∼2.30USD.
TitleSuppressedDuetoExcessiveLength 5
2 Methodology
Identifying speculativebubbles isa hardtaskevenindatasets withlongtime
series. Recently, based on previous works of Shiller [42], Mikhed and Zemcik
[44]regardedahouseasaninvestmentassetandusedastandardpresent-value
formula to derive implications for the relationship between house prices and
the cash flow associated with owning a property (rent).
Thefundamentalpriceisderivedfromthestandardnoarbitragecondition:
Et[Rt+1+Pt+1]
P = (1)
t
1+r
whereP isthepropertypriceindexatperiodt,E [.]denotesthemathematical
t t
expectation conditional on information at time t, R is the rent, and r is a
t
constant risk-free discount rate. Since the formula above holds for all t, the
property price index for the time t+1 is given by
Et[Rt+2+Pt+2]
Pt+1 = . (2)
1+r
Therefore, the Equation 1, by repeated forward iteration, can be written
as
Rt+1 Rt+2 Rt+k Pt+k
P =E + +...+ + . (3)
t t(cid:20)1+r (1+r)2 (1+r)k (1+r)k(cid:21)
Solving the Equation 3 yields the fundamental price:
∞
1
Ptf = (1+r)jEt[Rt+j] (4)
Xj=1
also called price reflecting fundamentals. This equation means that the fun-
damental price contain all expected future rents. In absence of a bubble, one
has the no-bubble condition [44]:
Et[Pt+k]
lim =0. (5)
k→∞ (1+r)k
This yields that the unique solution of Equation 1 is P = Pf under the
t t
hypothesis of non existence of a bubble.
The spread S between the house price and rent can be defined [41,44] as
t
1
S ≡P − R , (6)
t t t
r
which, based in the no-bubbles condition (Equation 5), can be rewritten as
∞
1 ∆Rt+j+1 1
St = rEt (1+r)j = rEt[∆Pt+1] (7)
Xj=1
 
6 MarceloM.deOliveira,AlexandreC.L.Almeida
where ∆Rt+j+1 = Rt+j+1 −Rt+j and ∆Pt+1 = Pt+1 −Pt. Note that the
stationarity of S implies the series {P /R } is stationary in the absence of a
t t t
speculative bubble, since P /R =1/r if S =0.
t t t
In order to estimate the fundamental value of property prices, we will
employ the price-rent ratio, PR. The PR ratio is defined as the average cost
of ownership divided by the estimated rent that would be paid if renting:
house price
PR= . (8)
annual rent
The PRratiofollows the conceptof stocks’price-earningsratio (PER),which
is defined as the ratio of the price of a share to the annual earnings of a
company in the current year. The PER ratio contains information about if a
given stock is over (or under) valuated [41,42]. Analogously, rents, as well as
corporateandpersonalincomes,areusuallyconnectedveryclosetosupplyand
demandfundamentals.Thisisthecauseonerarelyseesanunsustainable“rent
bubble” or an unsustainable “income bubble”. Therefore, a rapid increase of
housing prices combined with a flat or slow-increasing renting market can be
a signal of the beginning of a bubble [43]. As mentioned previously, in the
absence of bubble the series PR is stationary. The way to ascertain whether
t
ornotbubbles existis testing the stationarityofthe houseprice-to-rentratio.
This leads us to seek methods to determine whether series are stationary or
not.
Unit root tests are commonly used to determine whether a time series is
stationaryby using an autoregressivemodel. In its simplest form, the Dickey-
Fuller test[22]estimatesthe followingfirstorderautoregressiveAR(1)regres-
sion equation:
∆pt =α+(β)pt−1+ǫt,ǫt ∼iid(0,σ2). (9)
where p is the real price of the asset, α is the drift and β is the coefficient
t
of the model. The error term ǫ is an uncorrelated white noise process. The
t
DFtestcomparesthet-statisticsofresidualswithDFcriticalvalues.Thenull
hypothesis of the test is H0 : β = 0 which represents unit root versus the
left-tailed alternative hypothesis H1 : β < 0 stable root. (If the residuals in
first orderautoregressivemodel are still correlatedthe test canbe augmented
by ∆P for higher level autoregressive processes).
t−i
Incontrast,thesupremumaugmentedDF(SADF)testproposedbyPhillips
and Yu [21] is a right-sided test. The basic fundamental of this test is using
recursiveregressiontechniquestotesttheunitroot.InthecontextofDFtest,
the test is based on the follow regression:
∆pt =α+(β−1)pt−1+ǫt,ǫt ∼iid(0,σ2). (10)
The null hypothesis still is unit root behavior, H0 : β = 1 , but the al-
ternative hypothesis is explosive behavior, H1 : β > 1. The right-tailed ADF
statistics is computed in multiple recursive regressions for each sub-sample
whichstartwiththeinitialobservationbutthelastpointvaries.Letr1 andr2
be respectively the fractional starting and ending point of each sample. The
TitleSuppressedDuetoExcessiveLength 7
sample window rw = r2−r1 therefore varies from the initial size window r0
to the total sample.
The SADF statistic is based on the supremum value of the ADF statistics
obtained,
SADF(r0)= sup ADF0r2. (11)
r0≤r2≤1
The explosiveness of the process is|t{ezs}ted by comparing with the right-
tailed critical values of its limit distribution which is given by
r2WdW
sup ADFr2 → sup 0 (12)
0 Rr2W2dW
0
r0≤r2≤1 r0≤r2≤1R
where W is a standar|d{zW}iener process a|n{zd}→ denotes convergence in distri-
bution.
Instead of fixing the starting point of the sample, the generalized SADF
(GSADF)testextendsthesamplesequencebychangingboththestartingand
the ending point of the sample, implementing the right-tailed unit root test
repeatedly on a forwardexpanding sample sequence:
GSADF(r0)= sup ADFrr12. (13)
r0≤r2≤1
0≤r|1≤{zr}2−r0
TheGSADF statistics canbe definedasthe largestADFstatistic overthe
feasible ranges of r1 and r2. It is then used to detect the presence of at least
one bubble in the whole sample. Phillips et al.(2012) [23] demonstrate that
themovingsampleGSADFdiagnosticoutperformstheSADFtestindetecting
explosivebehaviorinmultiplebubbleepisodes,andworkswelleveninmodest
sample sizes.
In order to estimate the beginning and collapse dates of every bubble,
Phillipset al.(2012)suggestusingbackwardexpandingsamplesequences.Let
the fractional ending point fixed at r2 with the starting point r1 moving in
the range 0≤r1 ≤r2−r0. These ADF statistic sequences are denoted by
BADFr2 . (14)
r1 0≤r1≤r2−r0
(cid:8) (cid:9)
Hence the backward SADF statistic is defined as the sup value of the ADF
statistic sequence:
BSADFr2(r0)= sup BADFrr12 . (15)
(cid:8) (cid:9)
0≤r1≤r2−r0
The beginning (end) date of a bubb|l{ez}corresponds to the first date whose
the BSADF statistic becomes greater (smaller) than the critical values, esti-
mated by Monte Carlo simulation. (These critical values are calculated from
respective empirical distributions of each statistic, generated under the null
hypothesis (Equation 10 with β =1) )
8 MarceloM.deOliveira,AlexandreC.L.Almeida
3 Brazilian residential real estate market
Prior to the econometric analysis, let us briefly describe the data set used.
Untilrecently,Brazilhadnoreliableindicatorsfollowingthe pricebehaviorof
residential properties. For this purpose, since 2008, Fipe (Brazilian Institute
of Economic Research) developed an index, named Fipe-Zap, which use real
estateadsasasourceofinformation.Themajordrawbackwiththissourceisa
possible gapbetween the advertised price and the realized price. Nonetheless,
ifoneassumesthatbothpriceshaveasimilartrend,atleastinthemediumand
long run, an index for the advertised prices could be regarded as reliable real
estate market information. The index is based on a database of approximate
200 thousands real estate ads per month, and its methodology is detailed in
[45]. Figure 1 shows the price index for the seven biggest Brazilian cities (for
each index it was set an arbitrary value of 100 on August 2010). Almost all
the indices show high increases during the period, revealing that Brazilian
properties prices rose rapidly in the last few years. Unfortunately, only the
data for the two major cities, Sa˜o Paulo (SP) and Rio de Janeiro (RJ), is
available since December 2007. The sample data is also larger for these both
cities, therefore for a more precise statistical analysis, from now on we will
concentrate our analysis on the SP and RJ real-estate market.
200

São Paulo
Rio de Janeiro
Belo Horizonte
Recife
150 Fortaleza
Brasília
Salvador
100
60

Jan/08 Jan/09 Jan/10 Jan/11 Jan/12 Jan/13 Nov/13
Fig. 1 Fipe-ZapindexforthesevenbiggestBraziliancities.(Coloronline).
Since Fipe also calculates the monthly rental yield from its rent and sale
ads database, it is straightforward to obtain the PR ratio as the reciprocal
of the annualy rental yield. The resulting time-series with the PR ratio for
SP and RJ is shown in Fig. 2. In SP, we observe that the PR ratio grows
from around 11 to 18, and in RJ from around 15 to 22. This is equivalent
TitleSuppressedDuetoExcessiveLength 9
to a increase of approximately 64% and 47%, respectively. It should be noted
that a rising PR ratio is only a necessary but not a sufficient condition for
speculative misalignment from fundamentals.
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Rio de Janeiro
22 São Paulo
20
o18
ati
R r
P16
14
12

2008 2009 2010 2011 2012 2013 2014
Fig. 2 Price-rentratioforS˜aoPauloandRiodeJaneiro.(Coloronline).
4 Empirical Results
In order to test whether the movement of house prices in the Brazilian real-
estate market reflects deviation from levels supported by fundamentals we
performed more detailed analysis of the data shown in Fig. 2. Thus, we ap-
plied the recursive SADF and GSADF tests to the PR dataset. In our anal-
ysis, we choose the minimal window size r = 12 which ensures that there
w
are enough observations for initial estimation. The finite sample critical val-
ues were obtained via Monte Carlo simulations with 10,000 iterations. The
resultingSADFandGSADFstatisticsforthe RioandSa˜oPauloPRratioare
shown in Table 1. The SADF statistics indicate the existence of at least one
speculative realestate bubble in Rio de Janeiroduring the period with a 95%
confidence interval and in Sa˜o Paulo with a 99% confidence interval. Similar
conclusions canbe drawnfrom the GSADF test statistics, with bubble in Rio
de Janeiro (at 90% level) and Sa˜o Paulo (at 99% level).
Figures 3 and 4 show the results of the recursive SADF and GSADF tests
for Rio de Janeiro, respectively. The associated critical values are shown as
10 MarceloM.deOliveira,AlexandreC.L.Almeida
Table1 SADFandGASDFstatisticsforPRratioseriesandrespective90%,95%and99%
criticalvalues.
PRratio SADFt-statistics 90c.v. 95c.v. 99c.v.
RiodeJaneiro 1.3203 1.0106 1.3144 1.9812
S˜aoPaulo 2.6529 1.0106 1.3144 1.9812
GSADFt-statistics 90c.v. 95c.v. 99c.v.
RiodeJaneiro 1.9180 1.8312 2.1804 2.9606
S˜aoPaulo 4.1353 1.8312 2.1804 2.9606
1.5
1
0.5
0
−0.5
−1
Backward ADF sequence
−1.5 90% critical value sequence
95% critical value sequence
−2

2009 2010 2011 2012 2013 2014
Fig.3 RecursivecalculationoftheSADFtestforRiodeJaneiro.Thedotedlinesrepresents
the95%andthedashed90%criticalvaluesequences (Coloronline).
doted (95% level) and dashed (90% level) curves. The SADF test shows a
statistically significant explosive period from mid-2010 to the end of 2012.
The GSADF test identify a bubble beginning in mid-2010 from mid-2012.
Figures 5 and 6 show the results of the recursive SADF and GSADF tests
for Sa˜o Paulo, respectively. As before, doted (95% level) and dashed (90%
level) curves represent the associated critical values. In this case, the SADF
testshowsauniqueandlongexplosivebehaviorstartinginmid-2009.Onother
handtheGSADFtestissuccessfulinidentifyingmultiplebubbleperiods,with
the last one beginning in mid-2011.
The overall picture of Brazilian house price valuation provided by Figs.
3,4, 5 and 6 corroborate the existence of rational speculative bubbles in its
twomajorcities.Itisalsonoticeablethatthisconfirmsthepreliminaryresults
from glancing at Figure 2. On the other hand, it is needed to emphasize that
price-to-rent indices have obvious disadvantages and shortcomings. Although
the indices provide information about the dynamics of the price-to-rent ratio
over time, it can not provide information about the actual level of the price-
to-rent ratio. Analysis of other indicators such as a household income index